Crystallography & Symmetry
patterns derived from their groups
Concept

Displacement mode — where it appears

A pattern of atomic displacements transforming as a single representation of the crystal's group. Which patterns a set of atoms can supply is decided by their Wyckoff positions, and a multiplicity of zero is a distortion those atoms cannot make.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

p4m: which distortions the atoms of each position can make. Every Wyckoff position of p4m, with the number of independent displacement patterns of each symmetry its atoms supply. A zero is a distortion those atoms cannot make however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The row sums, weighted by the dimensions along the top, come to twice the number of atoms in the cell, which is the check that nothing has been lost. The last column is the number of free coordinates the position has, and it equals the multiplicity of the identity representation in that row: a displacement that keeps every symmetry is exactly a move of the position within its own Wyckoff set.

Which modes a site can carry

An atom on a mirror cannot move in a way that breaks the mirror while its images move with it: the displacements of a Wyckoff orbit carry a representation, and some of its pieces have multiplicity zero. The count of those pieces is a character, and the one that breaks nothing is the position's own freedom.

operations · Fundamental domain p6m: freezing Γ2 leaves p31m. The same crystal three times. On the left, a pattern with the full symmetry of p6m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is p31m, of index 2 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical.

An order parameter is a representation

The quantity that measures a lost symmetry is not a number the physics chooses freely: the parent group mixes its components, so it carries a representation, and the symmetry that survives is what leaves its value alone. Every prediction here is checked by moving the atoms and asking the detector.

space-groups · Landau

Named alongside it

The objects these essays reach for when they reach for this one.

CharacterEquivarianceFree parameterIsotropy subgroupKernelLandau conditionMechanical representationOrbit-stabiliserOrder parameterPhase transitionSite symmetrySymmetry breaking

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